Partial Differential Equations/The Malgrange-Ehrenpreis theorem

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Vandermonde's matrix[edit | edit source]

Definition 10.1:

Let and let . Then the Vandermonde matrix associated to is defined to be the matrix


For pairwise different (i. e. for ) matrix is invertible, as the following theorem proves:

Theorem 10.2:

Let be the Vandermonde matrix associated to the pairwise different points . Then the matrix whose -th entry is given by

is the inverse matrix of .


We prove that , where is the identity matrix.

Let . We first note that, by direct multiplication,


Therefore, if is the -th entry of the matrix , then by the definition of matrix multiplication


The Malgrange-Ehrenpreis theorem[edit | edit source]

Lemma 10.3:

Let be pairwise different. The solution to the equation

is given by

, .


We multiply both sides of the equation by on the left, where is as in theorem 10.2, and since is the inverse of


we end up with the equation


Calculating the last expression directly leads to the desired formula.

Exercises[edit | edit source]

Sources[edit | edit source]